# Ethylene frontier orbitals with PBE and B3LYP

Ethylene was built from the SMILES string `C=C` and optimized once with ADF at PBE/DZP. Two single-point calculations, PBE/DZP and B3LYP/DZP, then used that same geometry. Both calculations used an all-electron DZP basis and Normal numerical quality. Orbital energies are Kohn-Sham eigenvalues in eV.

## Optimized geometry

The optimized C=C distance is 1.3320 Å.

![Optimized ethylene geometry](figures/optimized_ethylene.png)

## Frontier orbital energies

| Method    |   HOMO (eV) |   LUMO (eV) |   Gap (eV) |
|:----------|------------:|------------:|-----------:|
| PBE/DZP   |     -7.0061 |     -1.1879 |     5.8182 |
| B3LYP/DZP |     -7.7749 |     -0.2106 |     7.5642 |

The nearest three occupied and three virtual orbitals are:

| Method    | Orbital   |   Energy (eV) |   Occupation |
|:----------|:----------|--------------:|-------------:|
| PBE/DZP   | HOMO-2    |      -10.4784 |          2.0 |
| PBE/DZP   | HOMO-1    |       -8.7810 |          2.0 |
| PBE/DZP   | HOMO      |       -7.0061 |          2.0 |
| PBE/DZP   | LUMO      |       -1.1879 |          0.0 |
| PBE/DZP   | LUMO+1    |        0.8367 |          0.0 |
| PBE/DZP   | LUMO+2    |        1.9669 |          0.0 |
| B3LYP/DZP | HOMO-2    |      -11.7766 |          2.0 |
| B3LYP/DZP | HOMO-1    |       -9.9698 |          2.0 |
| B3LYP/DZP | HOMO      |       -7.7749 |          2.0 |
| B3LYP/DZP | LUMO      |       -0.2106 |          0.0 |
| B3LYP/DZP | LUMO+1    |        1.3945 |          0.0 |
| B3LYP/DZP | LUMO+2    |        2.5806 |          0.0 |

![Frontier orbital energy-level diagram](figures/frontier_energy_levels.png)

## HOMO and LUMO isosurfaces

The four orbital pictures below were generated by `amsreport` from the corresponding ADF engine result files. Contrasting colors mark opposite orbital phases.

| Method | HOMO | LUMO |
|---|---|---|
| PBE/DZP | ![PBE HOMO](figures/amsreport_pbe_homo/report.jpgs/0.jpg) | ![PBE LUMO](figures/amsreport_pbe_lumo/report.jpgs/0.jpg) |
| B3LYP/DZP | ![B3LYP HOMO](figures/amsreport_b3lyp_homo/report.jpgs/0.jpg) | ![B3LYP LUMO](figures/amsreport_b3lyp_lumo/report.jpgs/0.jpg) |

## Conclusion

At this fixed PBE geometry, replacing PBE with B3LYP changes the Kohn-Sham HOMO-LUMO gap by +1.7460 eV. The HOMO and LUMO isosurfaces allow the comparison to distinguish an energy shift from a qualitative change in orbital shape. These Kohn-Sham gaps are method-dependent orbital-energy differences, not optical excitation energies.

## Calculation inputs

### PBE/DZP geometry optimization

```ams
Task GeometryOptimization

System
  Atoms
              C       0.6644850805       0.0279879686      -0.0236852009
              C      -0.6644849893      -0.0279879318       0.0236851499
              H       1.2534332310      -0.8786144292       0.0702991539
              H       1.1670384263       0.9805640030      -0.1565753856
              H      -1.2534332920       0.8786143859      -0.0702991171
              H      -1.1670384566      -0.9805639965       0.1565753997
  End
  BondOrders
     1 2 2.0
     1 3 1.0
     1 4 1.0
     2 5 1.0
     2 6 1.0
  End
End

Engine adf
  Basis
    Core None
    Type DZP
  End
  NumericalQuality Normal
  SCF
    Iterations 100
  End
  XC
    GGA PBE
  End
EndEngine
```

### PBE/DZP single point

```ams
Task SinglePoint

System
  Atoms
              C       0.6649813115       0.0278988522      -0.0242986367
              C      -0.6649810155      -0.0278982997       0.0243008447
              H       1.2789279657      -0.8699226278       0.0686049752
              H       1.1932069166       0.9740820859      -0.1564006741
              H      -1.2789292580       0.8699198605      -0.0686220517
              H      -1.1932059203      -0.9740798711       0.1564155425
  End
  BondOrders
     1 2 2.0
     1 3 1.0
     1 4 1.0
     2 5 1.0
     2 6 1.0
  End
End

Engine adf
  Basis
    Core None
    Type DZP
  End
  NumericalQuality Normal
  SCF
    Iterations 100
  End
  XC
    GGA PBE
  End
EndEngine
```

### B3LYP/DZP single point

```ams
Task SinglePoint

System
  Atoms
              C       0.6649813115       0.0278988522      -0.0242986367
              C      -0.6649810155      -0.0278982997       0.0243008447
              H       1.2789279657      -0.8699226278       0.0686049752
              H       1.1932069166       0.9740820859      -0.1564006741
              H      -1.2789292580       0.8699198605      -0.0686220517
              H      -1.1932059203      -0.9740798711       0.1564155425
  End
  BondOrders
     1 2 2.0
     1 3 1.0
     1 4 1.0
     2 5 1.0
     2 6 1.0
  End
End

Engine adf
  Basis
    Core None
    Type DZP
  End
  NumericalQuality Normal
  SCF
    Iterations 100
  End
  XC
    Hybrid B3LYP
  End
EndEngine
```